Let's outsource our prisoners to Mexico

I have been reading about the overcrowded conditions in the California prisons, and that there may be an early release of prisoners. The perfect solution is to pay Mexico to build prisons, then California can send all of the prisoners who are in the U.S. illegally to these Mexican prisons. Of course, we would pay Mexico to house the prisoners. We outsource everything else to Mexico, so why not these prisoners? I know that it would cost us much less than it costs to build prisons and house the prisoners here. This would cut down the cost to house, feed and clothe people who aren't even supposed to be in this country. It may also make illegal aliens think twice about breaking the law if they think they will be sent to prison in Mexico where it isn't so pleasant as in California.